1. Field of the Invention
The present invention relates to methods in the field of digital imaging, and more particularly, to methods for computer-assisted shading and rendering of three-dimensional surfaces.
2. Description of Related Art
A core aspect of computer-assisted rendering involves calculating reflected light rays resulting from a defined modeled light source, based on the surface geometry of an object to be rendered, and the reflectance properties of its surface such as may be caused by surface properties at a microscopic scale. Traditionally, fairly simple theoretical models, such as the Phong model, have been used to model microscopic reflectance properties. But many common materials, such as fabrics, cannot be rendered in a photo-realistic way using simplified models.
A more sophisticated approach to modeling reflectance properties entails the use a function that defines reflectance as a function of the angle of incidence and angle of reflectance. This function is usually referred to as the Bidirectional Reflectance Distribution Function (BRDF), which may be expressed as
                                          ρ            ⁡                          (                                                θ                  i                                ,                                  ϕ                  i                                ,                                  θ                  r                                ,                                  ϕ                  r                                            )                                =                                    ∂                                                L                  r                                ⁡                                  (                                                            θ                      r                                        ,                                          ϕ                      r                                                        )                                                                    ∂                                                E                  i                                ⁡                                  (                                                            θ                      i                                        ,                                          ϕ                      i                                                        )                                                                    ,                            (                  Eq          .                                          ⁢          1                )            where E is the irradiance (incident flux) and L is the reflected radiance (reflected flux). The angle of incidence is defined by θi,φi, and the angle of reflectance by θr,φr. The symbols 0 and φhere represent angular coordinates, for example, as used in a polar coordinate system. The BRDF may be generally described as a function that defines the light that will be reflected in every direction from a surface, for each incident ray angle. The BRDF ratio ρ(θi,φi,θr,φr) for a defined pair of incident and reflected rays is referred to herein as a “BRDF value.” Materials that exhibit anisotropic reflectance, such as fabrics, are difficult to model theoretically. Sufficiently accurate theoretical models generally are not capable of sufficiently photo-realistic rendering in demanding environments, such as motion-picture production, particularly for applications that require seamless compositing of digitally rendered and photographic images.
As an alternative to modeling a surface's BRDF, a sample of actual material may be measured using a measuring apparatus. Various measuring apparatus are known in the art, including physical gonioreflectometers and imaging gonioreflectometers. For rendering applications, using measured BRDF data is subject to several limitations. For one, traditional measurement apparatus often does not yield reliable or reproducible BRDF measurements. Thus, the final rendered result may differ noticeably from a photographic image of the same material. One of the reasons for photo-realistic rendering is to provide digitally rendered images that can be seamlessly inserted into photographic images, for the purpose of creating digital special effects. Inaccurate rendering of a material, such as may be caused by insufficiently precise measurement of the material's BRDF, will inevitably frustrate this purpose.
Second, even if sufficiently accurate BRDF data can be gathered using a carefully designed reflectometer, the data will only be sampled over a discrete number of points of a sample surface. Because the collection of BRDF data for each point of the sample surface is inherently time-consuming, available BRDF data for a given material tends to comprise a sparsely-sampled data set. Merely interpolating between sparsely-sampled data points may not provide a sufficiently accurate model of the actual BRDF to achieve a photo-realistic result.
These limitations have hindered the application of photo-realistic rendering, particularly for demanding production environments such as motion picture production. It is desirable, therefore, to provide a method for overcoming them.